Mirror Fugue/euguF rorriM

A while back I mentioned a college textbook on counterpoint: Modal and Tonal Counterpoint: From Josquin to Stravinsky by Harold Owen. It is intended for undergraduates, obviously. Just now I tried to look up "mirror fugues" to see what he had to say and there was nothing. Mirror fugue was for me the occasion of one of those stunning moments of realization that one has in one's life. Let me set the stage.


By a quirk of fate I never took a single class of counterpoint in my undergraduate years. I transferred from one university to another after first year theory. In the first university, they spent first year on harmony, reserving second year for counterpoint. In the second university, they did it the other way around. When I applied to the second university (McGill in Montreal), they told me I had to do some placement tests to see exactly what they would give me credit for and what I would have to enroll in. One of the placement tests involved counterpoint, so I spent the summer teaching myself the basics. As a result, I passed the placement and did not have to do the counterpoint course. I only ever did one counterpoint course, Fugue, the most advanced of all, and I didn't do it until I was in the doctoral program, many years later. It was in this course that I had one of those stunning moments of realization.


Fugue involves a number of interesting techniques, some of which involve transforming the subject. You can make the subject twice as long, by doubling the note values, (quarter notes become half notes and so on) or by inverting the subject: a step up becomes a step down and a leap down becomes a leap up and so on. Bach has lots of great examples. This one, from the Art of Fugue, manages to combine three different versions of the subject in the first three measures. First, there is the subject, which is itself a transformation of the basic subject of the whole Art of Fugue. Then, in measure two the subject again, but upside down and in double note values. Finally, in measure three, the subject in the original note values, but upside down. I omitted the continuation of the first, tenor, voice for clarity.

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Just a bit later, in measure five, the bass enters with the subject upside down and in quadruple note values with the quarter note becoming a whole note! Let's hear that whole piece, the Contrapunctus VII:

Another technique is called "invertible counterpoint" and involves putting the upper voice below the lower, or vice versa. The problem is that not everything works upside down. When you invert voices at the octave, the most obvious interval, the intervals change. Octaves become unisons, 7ths  become 2nds, 6ths become thirds, and so on. The problem, oddly enough, is with the 5ths, which become 4ths. The dissonant intervals like the 7th, stay dissonant in inversion, but the 5th, a consonant inverval, becomes the 4th, which is usually a dissonant interval needing resolution. So just avoid 5ths! Heh. You can invert at other intervals, such as the 10th, but they are more difficult because parallel thirds and tenths become parallel unisons or octaves and therefore must be avoided.


The kind of counterpoint that is written so that it may be inverted is called either invertible counterpoint, or double counterpoint. Now imagine if someone were to write an entire fugue, in three or four voices, and wrote it so that it could all be inverted. Almost unimaginable. But Bach did it three times in the Art of Fugue. Once in three voices and twice with four voices! Here, chosen somewhat at random, are three measures from Contrapunctus XVIII (in my score: the recorded versions call this Contrapunctus 12a and 12b). The top staff is from the right side up version, and the bottom staff is the mirror image:

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As you can see, the soprano becomes the bass, the alto the tenor, the tenor the alto and the bass the soprano. And everything works! I think you have to try writing invertible counterpoint a bit before you can appreciate how difficult this is. Put it this way, in the Fugue course we had been struggling with invertible counterpoint for a few weeks. And then the professor said, "oh yes, and Bach wrote some fugues in which the whole thing, all four voices, were invertible." I nearly fell off my chair. So let's hear the two versions of the fugue. First, right side up. The brief excerpt above occurs about 50 seconds in:

Then, upside down or "inversus":

The whole idea of writing a whole fugue so as to be invertible is at the very stratosphere of counterpoint. I can't think of a single composer who has managed it since Bach...

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Shostakovich and Fugue

A fairly recent textbook on counterpoint is Modal and Tonal Counterpoint: From Josquin to Stravinsky by Harold Owen. In the realm of the textbook, the modernist ideology still is largely unquestioned. The story goes that by the end of the 19th century tonality was moribund and music had to be rescued by an entirely new approach. This was provided by Arnold Schoenberg, Bela Bartok and Igor Stravinsky, among others. Schoenberg emancipated the dissonance and developed atonal serial composition while Bartok explored incorporating folk melodies and rhythms and Stravinsky adopted a variety of approaches including neo-classicism (using the manner of earlier music), polyrhythms and rhythmic cells and other techniques like octave displacement and polyharmonies. Theorists tend to be naive progressivists as they regard all technical advances as worthy. The corollary to that is that everything that is not a technical advance is unworthy.

UPDATE: What I intended to mention here, is that there is absolutely no mention of Shostakovich in the counterpoint text, which peters out rather unconvincingly with a little Hindemith, a piece from Mikrokosmos by Bartok and the Great Chorale by Stravinsky--surely, all of them rather trivial examples of counterpoint when set beside Shostakovich?

So due to this litany, theorists and historians have tended to have a blind spot and smack in the center of it is Shostakovich. He was parodied by Bartok and ridiculed by most composers in the main stream of 20th century practice. Why? He didn't follow the correct technical progression and was therefore seen as an apostate (or hack). This was often attributed to the political repression in the Soviet Union that forced him to write music conforming to 'socialist realism' with its insistence on the use of folk song, the avoidance of 'formalism' (a word for the influence of Western avant-garde music) and the creation of an heroic effect.

Now an argument could be made that some of Shostakovich's music, the more public genres such as symphony, for example, do show a superficial conformity with the demands of socialist realism. Discussion has raged around this question and I don't want to take it up here. A good locus classicus for it is the article by Richard Taruskin "Public lies and unspeakable truth: interpreting Shostakovich's Fifth Symphony" in Shostakovich Studies, edited by David Fanning.

The Twenty-Four Preludes and Fugues for piano, op 87, composed in 1950-51, did raise some opposition as they seemed to many to be a bit outside the path that Shostakovich was supposed to be following and it took a masterful performance by the dedicatee, Tatyana Nikolayeva, for the Committee for Artistic Affairs to give authorization for publication.

These pieces, the first successful attempt since Bach to write preludes and fugues in all the keys, were the fruits of a Bach competition that Shostakovich was present for in Leipzig, celebrating the 200th anniversary of Bach's death. Prior to this he had not been particularly disposed to the fugue genre, but he ended up writing a prelude or a fugue every few days for the next couple of months.

The result was a work of real racination, in my neologism. I talk about that here. Sometimes in music history, composers find themselves in a situation where the only solution seems to be to go back to some fundamental spring of music, some bedrock, and start all over. This happened around 1600 and again around 1970. I think that Shostakovich is doing something similar here--rediscovering fundamental things about counterpoint and harmony. Let's look at the first fugue in C major. It is so totally in C major that there is not a single accidental in the entire 106 measures. Here is Nikolayeva playing the Prelude and Fugue in C major. The Fugue starts around 2:31:

This sounds nothing like a Bach fugue, of course. For one thing, it is as much modal as tonal. When Shostakovich has an entry on E, he makes no attempt to inflect to E minor. Similarly, when he makes an entry of the subject on B, it results in the Lydian mode with the perfect fifth of the subject becoming a diminished fifth [UPDATE: Lydian mode would give you an augmented 4th. This is actually in Locrian mode, but who has ever heard of Locrian mode?]. Despite the adherence to fugue style, it comes out as Shostakovich. A piece of music like this is a repudiation of the unspoken assumptions of the 20th century avant-garde which we might summarize as the following: thou shalt not write tonal music, or if you do, it has to be a 'wrong-note' parody like Stravinsky's Great Chorale from L'Histoire du soldat; thou shalt not attempt to revive outmoded musical forms, again, except as parody; thou shalt develop thine own unique style with as little reference to other kinds of music as possible and finally, thou shalt move through different periods in thy work so as to seem fashionable.

Op 87 by Shostakovich is about as far from these strictures as possible. Perhaps this is why the work was not taken seriously for a long time. It does take some listening. At first I heard little of interest there. But, like the Bach Art of Fugue it grows and grows on you. The preludes are a diverse collection of musical styles. The one in B minor is like a Baroque overture:

But the fugue that follows is odd. The subject is stated in octaves, but then another subject, or countersubject, is stated separately. Then they join together. That second subject provides much of the development with the first coming in occasionally, sometimes in stretto. The harmonic layout also is like nothing Bach would have done with entries in G minor and C minor. The one thing that seems clear is that this is a serious piece of music, a profound meditation...

I haven't put up an analysis of these pieces as I often do with Bach. In fact, outside of a few Russian theorists, I doubt anyone has analyzed this music! But more and more people are beginning to appreciate op 87 and I have even seen one claim that this is the greatest piano work of the 20th century. You know, that wouldn't surprise me a bit.

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How to Build a Fugue

A friend of mine went through university as a fine arts major. It was in the 70s so stuff like life-drawing and all those silly traditional things were down-played as being pointless in the brave new world of modernism (or was it post-modernism already?). Instead they spent their time doing 'concept art'. He's still pissed off about it because as the years went by he realized how important it was to have learned the craft, the technique of art.

It was a bit like that in music, too, though not so extreme. All music students always learn basic theory and history, though not all schools take it very far. The young composers in one of the universities I attended seemed remarkably free of any association with the traditional craft of composition. I remember one concert that was a 'happening' where one of the performers fried pork chops while another sang cabaret songs. Ah, well...

But the truth of it is, as T. S. Eliot once remarked, no vers is libre to the poet who wants to do a good job. One of the most fundamental problems of composition is the melding of variety and unity. The great composers continue to astound us with how they achieve this. Let me send you back again to this post where I put up Bach's Goldberg Variations and Beethoven's Diabelli Variations. Bach did thirty variations on a simple air, every third variation of which was a canon at a different interval: unison, second, third and so on. The Beethoven is thirty-three variations on a silly waltz by Diabelli. Go have a listen, if you have time. My experience is that the wonder at what they did grows with each listening.

Variation form is just one way of unifying a piece. Other techniques have been developed. One of the most interesting is the fugue. You can't really speak about the form of a fugue the way you can of the form of a minuet or the form of a rondo as each fugue is really in a different form. What makes a fugue a fugue is not the form, but the texture. In a fugue a subject is presented which is answered in another voice, usually at the fifth. Typically all the voices enter with the same subject. Following this is an episode where elements of the subject and/or the countersubject (what the other voices are doing when one voice is presenting the subject) are varied and developed. In a normal entry, the second voice waits until the first is finished before presenting the subject, but another possibility is the stretto, where the subject is stacked up on itself. So, lots of unity in a fugue! The trick is to achieve variety.

Here is the subject and countersubject from the first fugue, in C major, from Bach's Well-Tempered Clavier, Bk 1:

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I said every fugue had a different form? This one is a good example. Most fugues have episodes of development between entries of the subject. This one doesn't. It has 24 entries of the one-and-a-half measure subject in a fugue only 27 measures long. How's that possible? The answer is stretto. He stacks them up. This is possibly the most strettoish fugue ever. Here is the first stretto:

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Stretto can cause problems of course: how do you know that stacking the subject on top of itself won't result in some nasty sounds like parallel fifths? Bach could apparently hear a subject and just know what strettos were possible. In any case, this one works fine a fourth below and a quarter note later. But there are a lot of strettos in this fugue. Here is another one:

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Starts with the same stretto, a fourth below and a quarter note later, but adds another a tenth below and three quarter notes later and yet another an octave plus a seventh below and five quarter notes later. In the actual fugue there are other voices playing as well. In order to show the stretto clearly I have left them out. Note that the last entry of the subject is irregular: a quarter note instead of an eighth rest followed by an eighth note. This is to avoid parallel octaves between the alto and bass. What Bach has done is layer together three more entries of the same subject before the first is finished!

I said all fugues were different? The very next one in the Well-Tempered Clavier, the C minor fugue, has no strettos whatsoever, but instead is a compendium of contrapuntal inversion. But that's a story for another day. Let's listen to the C major fugue. This is Friedrich Gulda playing both the prelude and the fugue, which starts around 2:09:

It is usually played a bit faster, but the slow tempo lets you hear everything very clearly.

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